| L(s) = 1 | − 3.09·3-s − 2.53·5-s + 6.56·9-s + 2.41·11-s − 0.891·13-s + 7.83·15-s + 1.35·17-s − 7.44·19-s + 7.55·23-s + 1.42·25-s − 11.0·27-s − 9.21·29-s + 7.97·31-s − 7.46·33-s − 3.88·37-s + 2.75·39-s + 41-s − 10.1·43-s − 16.6·45-s − 11.3·47-s − 4.17·51-s + 6.76·53-s − 6.11·55-s + 23.0·57-s − 5.15·59-s + 12.0·61-s + 2.26·65-s + ⋯ |
| L(s) = 1 | − 1.78·3-s − 1.13·5-s + 2.18·9-s + 0.727·11-s − 0.247·13-s + 2.02·15-s + 0.327·17-s − 1.70·19-s + 1.57·23-s + 0.284·25-s − 2.12·27-s − 1.71·29-s + 1.43·31-s − 1.30·33-s − 0.638·37-s + 0.441·39-s + 0.156·41-s − 1.54·43-s − 2.48·45-s − 1.64·47-s − 0.585·51-s + 0.929·53-s − 0.825·55-s + 3.04·57-s − 0.671·59-s + 1.53·61-s + 0.280·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 + 3.09T + 3T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 0.891T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 + 7.44T + 19T^{2} \) |
| 23 | \( 1 - 7.55T + 23T^{2} \) |
| 29 | \( 1 + 9.21T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 + 3.88T + 37T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 6.76T + 53T^{2} \) |
| 59 | \( 1 + 5.15T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 2.32T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 5.32T + 83T^{2} \) |
| 89 | \( 1 - 1.69T + 89T^{2} \) |
| 97 | \( 1 + 7.08T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.21323058502515862406845009084, −6.69913131756171523234550081910, −6.26016826683484399249776543205, −5.25564299209377504685210159270, −4.81132125967701785147756763527, −4.05796375895305955571316854580, −3.45379769490457157151798754890, −1.94969620358262216707352746851, −0.866522084531275348954307906042, 0,
0.866522084531275348954307906042, 1.94969620358262216707352746851, 3.45379769490457157151798754890, 4.05796375895305955571316854580, 4.81132125967701785147756763527, 5.25564299209377504685210159270, 6.26016826683484399249776543205, 6.69913131756171523234550081910, 7.21323058502515862406845009084
